Continuity of Drag and Domain Stability in the Low Mach Number Limits

Abstract

We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We assume that the Mach number is proportional to a small parameter ε and that the general boundary of the body contains small asperities of amplitude proportional to εα for a certain α > 0 and suppose the Navier’s slip condition on this rough boundary. We show that time averages of the drag functional converge, as ε → 0, to the corresponding time averages of the drag for the limit system, whereas the limit system is turning out to be the incompressible Navier–Stokes system with no-slip condition on the smooth limit body.

Mathematics Subject Classification (2010)

Primary 99Z99 Secondary 00A00

Communicated by R. Finn

The work of E. Feireisl was supported by Grant 201/09/0917 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503. The work of E. Friedmann was supported by Olympia-Morata-Programme of Heidelberg University.